On the Use of Series Expansions for Kirchhoff Diffractals

In recent years, some methods have been devised to evaluate the field scattered by natural surfaces modeled by using fractals. In particular, use of the Kirchhoff approximation allows expressing the field scattered by a fractal surface in terms of two different series expansions. In this paper, the practical applicability of these expansions is addressed. To this aim, we first of all reformulate the derivation of the two series in order to clearly identify the key parameters on which the series behavior depends. Then, we perform a rigorous analysis of the properties of the two series. Based on such an analysis, we present suitable truncation criteria which allow understanding how to practically employ the two series expansions to compute the scattered field with a controlled error. A deep analysis of the range of applicability of the presented truncation criteria is also included. This allows providing a criterion which, given the surface and illumination parameters, and given the required accuracy and the computer floating-point format, allows us to choose which of the two series, if any, can be used, and how it can be properly truncated. Based on the presented analysis, we verify that for values of surface parameters of practical interest and for which the Kirchhoff approach can be used, for reasonable values of the required accuracy, and if the IEEE standard floating-point double-precision numbering format is used, then there is always at least one of the two series that provides an approximation of the scattering integral with the required accuracy.